Bone constraints not working on first bone in chain
The issue was with the bone at the start of the chain being set as Connected. This is because the bone chain was made using the Skin modifier on a vertex chain. Normally if you add a bone and then extrude it to make a chain, the first bone is not Connected. Apparently the armatures made by Skin are. The Connected checkbox is greyed out, but still can be unchecked. Since the bone has no Parent, the Connected status was locking it in place.